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December 28, 2013

The Bus Stop Problems

Young Economics whippersnapper Evan Soltas posed two problems to do with Bayesian probability:

You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop as a Poisson process, with an unknown (to you) rate, λ\lambda. You don’t know λ\lambda, but say you have a prior probability distribution for it, p0(λ)p_0(\lambda).

What’s your expected wait time, ⟨T⟩\langle T\rangle, for the next bus to arrive?

Let’s add some more information. Say that riders arrive at the bus stop via an independent Poisson process with an (unknown to you) rate, μ\mu. Whenever a bus arrives, all those waiting at the stop get on it. Thus, the number of people waiting is the number who arrived since the last bus. Say you arrive at the stop to find nn people already waiting. You wait for a time, tt, at which point there are NN other people waiting at the stop (i.e., N−nN-n arrived while you were waiting).

These questions illustrate one of my favourite points of view on Bayes Theorem, namely that it induces a flow on the (infinite-dimensional!) space of probability distributions. Understanding the nature of that flow is, I think, the key task of the subject.

Infinite dimensions are hard to get an intuition for, so one of the first tasks is to cut the problem down to a finite-dimensional one.